Policies for Math Thought: Voting, Conflict and Power
Math 120
Spring 2002

Instructor: Janice Sklensky
Office Phone: (508)286-3973
Office: Science Center 109
Office Hours: See my schedule
E-mail: jsklensk@acunix.wheatonma.edu

Below, I discuss

Course Materials:
Mathematics and Politics: Strategy, Voting, Power and Proof, by Alan Taylor.

The text, and a calculator if you have one, should be brought to class every day.

Overview:
The more you look around you, the more you will discover that math is everywhere. This semester, we'll focus on politics as a (timely) forum for exploring mathematics.

We will use the concept of escalation to motivate a discussion of game-tree analysis. That will lead us into a discussion of Game Theory. In an article on the math in ``A Beautiful Mind'', the Boston Globe describes Game Theory as being used by economists, political analysts and business leaders to describe and influence everything from Wal-Mart's post-holiday sales to India and Pakistan's armed standoff. We will see how Game Theory can be used to get a feel for what happened during the US-Soviet arms race and the Cuban Missile Crisis, analyse some of our situations using Game Theory, and see some of the limitations of mathematical models.

The second half of the semester will focus on Voting Theory. We will learn about some of the intricacies of yes-no voting, and develop methods for measuring how much power individuals have in deciding yes-no questions when different individuals' votes are not all counted equally, such as when a partner in a business has veto power, or when different states have different numbers of electoral votes. We will conclude the semester by studying several different methods of voting, discuss whether the most common methods of determining the winner of an election are fair, and discuss the question of whether there even is a fair method of deciding a winner.

Through our focus on Game Theory and Voting Theory, I hope you will come to appreciate both the potential and the limitations of mathematical analysis, becoming a sophisticated and skeptical consumer of mathematical models. Throughout the course you will see some very beautiful mathematics.

This class requires no specific mathematical background (beyond arithmetic and some algebra), nor does it assume any background in political science (I myself have no political science background). Calculation is emphasized less, probably, than in most of your previous courses, and deductive reasoning is emphasized more.

I come to this class knowing that the class will consist of students with a variety of backgrounds. Some of you are starting the course with a very positive attitude towards mathematics, some may not have had the best mathematical experience, while still others of you have had just plain bad mathematical experiences. I try to establish an atmosphere that is challenging and engaging for all of you, but supportive and reassuring for those who need it. I want you all to succeed, and most of you will succeed if you follow the right strategy (see ``How can you succeed?'' and ``Is this the right class for you'' below). I think this material is really interesting, and hopefully you will too! For those who have had negative experiences in the past, or who are ``only taking this class because you have to'', keep an open mind!

As is true with all courses, how much you actually learn in Math Thought is entirely up to you. As you read through how the course is structured, you will see that a lot is expected of you. Your responsibilities for this class will consist of reading the textbook daily, doing many homework problems, working on a few projects, and a comprehensive final.

Is This the Right Course for You?:
The primary qualifications for this course are a willingness to work hard and an interest (a willingness to consider the possibility of being interested) in how math can be used to unravel some of the murky world of politics.

As I mentioned above, I do not assume very much in the way of mathematical background; all I ask is that if you have previously had negative mathematical experiences you put them aside and be open to the possibility that you might enjoy this class.

There are some people, however, for whom this is not the right class. Students whose choice of major and/or study abroad makes more than one math class during the course of their college career undesirable should think about their needs: those who might major or minor in psychology, sociology, biology, education, political science, or economics may be required (or recommended) to take other math courses.

People who want to learn techniques they'll use often (as opposed to learning some of the ways math relates to the world around you) should consider taking Statistics or Universal Machines.

Furthermore, while an active interest in politics is not necessary, it is necessary that you not be turned off by the occasional mention of the India-Pakiston armed standoff, the various conflicts between the US and the Soviet Union during the cold war, or the last presidential election. If you are such a person, you may want to consider taking the other Math Thought course being offered now (which is also on very interesting stuff) or taking one of the Math Thought courses being offered next year, which may focus on different subjects.

Finally, if you plan on not putting a lot of effort into this class, you should wait until you can put the effort into it: you will get more out of it (as well as being more likely to pass it), and it's more fair to the rest of the class.

How can you succeed in this course?:
As I mentioned above, being interested (or willing to be interested) and being willing to work hard are the two critical factors for success in this course.

Plan to spend an average of 9 hours a week outside of class working on this course. As usual, some weeks you will spend more time on this class, while others will be less frenetic (relatively speaking). Also, of course, be aware that some people can get away with spending less time on the course than others can. Just because your friend or neighbor isn't working hard doesn't mean that you won't have to.

Another key to success: admit it when a concept or connection is eluding you. Come see me in my office, and/or consult with friends.

And finally: think about forming study groups. As I discuss below, working alone has its value, but so does working in groups.

Office Hours :
Please come visit me! Chat about mathematical thoughts, or bring questions. If you do have questions, come right away, don't stew over them and let them evolve into a serious situation!

If you come during my office hours, you of course do not need to make an appointment beforehand -- just stop by! If you can't make my office hours, make an appointment with me for another time.

Contacting You:
It may come up that I need to e-mail the class about something: a change to the problem set, a clarification of a problem. Please make sure I have the e-mail address you use regularly, and please do check your e-mail regularly. You can also, of course, always e-mail me at jsklensk@wheatonma.edu

Reading technical material is an extremely valuable skill, and is becoming more necessary in all areas of our lives all the time. One of the goals of this class is that you become more comfortable reading mathematical prose.

Before each class meeting, I expect you to have read the material that we will be discussing that day. At first it may seem difficult, but stick with it -- as you get used to Taylor's writing style, and as you practice, you'll find it getting easier. The key to reading mathematics is (as you've probably heard before) to read it with pencil in hand. Whenever you come to an example, try to work it out yourself; fill in the missing details in the book. Really read the diagrams and try to figure out where they come from and what they have to do with the text.

Class Notes:
The notes you take in class will probably serve as the main text for the course, with Taylor's book forming the backbone. The book will give you an introduction to what I'll be doing, while your notes will have alternative explanations and extra examples.

It is essential to take clear and complete notes. If you miss a class, borrow someone else's notes, quickly photocopy them so you can return the originals, and then recopy the notes in your own notebook. The point of this, obviously, is in the process of recopying the notes, you will be thinking about and learning the material. Keep all your notes in the same notebook, rather than sprinkling them randomly among your various notebooks.

Problem Sets:
While the ideas presented in this class will (hopefully) make sense to you when you see me doing them, and even perhaps when practicing them yourself during class, there is of course a big difference between familiarity and mastery. As I'm sure you know, we all learn math by doing it. To encourage you to practice the concepts we are learning, and to give you credit for those efforts, I will be collecting weekly homework.

Furthermore, for most people learning math is not a solitary process. We learn best by thinking hard about something, and then trying to explain our thoughts. That is, learning math is best accomplished through a combination of group and individual efforts. For that reason, your weekly problem sets will alternate between being done individually and in groups.

The problem sets will usually consist of a mixture of ``drill'' problems, which are of course critical in making sure you understand the concepts we're learning, and some interesting but perhaps more time-consuming or conceptually more difficult problems.

I will say more about group problem sets as the first one approaches, but it is important that you know from the outset that you may not divide the problems for group problem sets up among the members of yoru group, and the groups should consist typically of only two people, although the occasional group of three may be accepted when necessary.

Problem Set Bureaucracy:
Your assignments will be posted on the web. The assignments can be found through links toward the bottom of the course web page.

Problem sets will be due every Friday by 2:00pm at the latest (you may certainly turn them in during class if you are ready to).

Begin the week's problems on Saturday -- they represent a week's worth of learning and so should be worked on throughout the week. I will usually assign more problems than can easily be done in one night, so don't put them off until Thursday.

I will set aside 5-10 minutes at the beginning of class Wednesdays and Fridays to answer questions on the homework.

Consult the Guidelines for Homework Presentation for information on how your problem sets should look.

 Late problem sets will not be accepted!
I will drop the lowest problem set score at the end of the semester. Those people whose problem sets are routinely good will be given some extra credit, however, to discourage people purposely blowing off one problem set.

Projects:
To give you an opportunity to get more of a feel for game theory and voting theory by applying them either to situations more relevant to you or to more complicated situations, you will work on at least four projects this term, in groups of two.

Most of the work you will do outside of class. The project consists not only of the mathematical solution to the situation, but (equally importantly) your description of the solution and why it is true.
Do I accept late projects?
Only in case of dire emergency! Because projects are worth a sizable chunk of your grade, I am reluctant to give all the members of a group a zero when the fault may have only been due to one person. However, unless the group has an irrefutable reason for turning in the project late, I take off significant points for each day the project is late. Even if your group does have a good reason, I may take off some points for each day the project is late, particularly as time goes on.

Exams:
On an exam, you can expect to see some combination of problems similar to homework problems, problems that apply the concepts we've studied to an unfamiliar situation, explanations of concepts, and occasionally proofs.

While last-minute cramming may be the study technique you've used in the past, or the technique you think most appropriate to a General Education class, it probably will not serve you well in this class. I suggest re-reading the notes, summarizing the important concepts and techniques while you go; redoing as many problems as you possibly can ( not merely reading the solutions you figured out at the time you turned in the problem set), and practicing writing proofs and explanations of concepts. By ``practice'', I do not mean memorization -- the words may be different every time. It takes practice explaining something yourself to really be able to explain a concept. So read through an explanation and work on it until you really feel you understand it. Take a break to give yourself time to forget the specific words. Then take a blank piece of paper and try to write out the explanation or proof yourself. Compare to the original and repeat until you've got it down pat!

Obviously, from the above discussion, studying for an exam in this class is not a small thing. You'll have to spread it out over several days, so be organized.

Exam Bureaucarcy:
I will give three exams, to make sure throughout the semester that you are learning the material. Each of these will be given on Monday evenings, and I will let you spend up to 2 hours on them (I will write them intending for it to take the ``average'' student about 1 hour, so that should give you plenty of time.)

These exams are tentatively scheduled for:

• Monday February 25
• Monday April 1
• Monday April 22
You will be allowed to bring one 8.5 x 11 sheet of paper, with handwritten (by you) notes, front only, to use during the exams and to turn in with the exams.

We will also, of course, have a cumulative final. This final will be a pre-scheduled exam, and will be given from 2 - 5pm on Tuesday May 7. I can not reschedule final exams.

If you can not take a midterm exam at the scheduled time, either because of scheduling conflict or due to illness, you must notify me in advance either by phone or by e-mail. If your reason for missing is acceptable, we will arrange that you take the exam early. Not beign prepared is not an adequate reason for missing an exam, of course. If you miss an exam without notifying me in advance, I reserve the right not to give you a make-up exam. I will not give any individual more than one make-up exam during the semester.

Attendance:
Clearly, missing class is not a wise idea. If you do miss class, it is of course your responsibility to find out any assignments, and to get a copy of the notes and of any hand-outs.

Evaluation:
I expect to use the weights below, although I reserve the right to change my mind if the semester does not go as expected.

 Problem Sets 23% Projects 12% Midterm Exams 45% Final Exam 20%
If you question the fairness of any grade, bring it to me within a week of receiving it.

Honor Code:
I expect you to abide by the Honor Code. If I have any reason to suspect that perhaps a violation has occured, I will ask the Judicial Board to investigate the matter. Below are some guidelines on what constitutes violations of the honor code in this class.

Problem Sets and Projects: You may work with anybody you want. You may use any references that help you figure out how to do the problem on your own; you may not use any references (people, old projects, books, the web, for instance) which tell you how to solve it or lead you to the solution. You must understand how to do every problem, and you must cite references if you've received assistance from any source. When doing group projects or group problem sets, you may not divide it into different parts--you must do them all together, and you must make sure every member of your group understands every part. Exams: You may not use any notes, books, or colleagues as reference during the midterms and final, except for your ``cheat sheet'', which must conform to my stated rules. You may not use a calculator unless I specify that you may, and you may not use a graphing calculator.

Janice Sklensky
Wheaton College
Department of Mathematics and Computer Science
Science Center, Room 109
Norton, Massachusetts 02766-0930
TEL (508) 286-3973
FAX (508) 285-8278
jsklensk@acunix.wheatonma.edu

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